Propagation of a normal mode in the parabolic approximation

Abstract

The parabolic approximation [Maury Center Non-Ray-Tracing Workshop (May 1973)] to the acoustic wave equation is studied for a case where the normal-mode solution is valid. Errors arising in the parabolic approximation are analyzed using a normal mode as the input. It is shown that two types of errors occur. The first is dependent on the choice of range increments and may be made arbitrarily small by proper choice of range steps. In contrast, the second type of error, a phase error, is inherent in the method and can be expected to be present in any practical situation. Results of a computer program confirming this analysis are presented. [Supported by Naval Ordnance Systems Command.]

Theoretical description of the parabolic approximation/Fourier split‐step method of representing electromagnetic propagation in the troposphere

“…The impact of the parabolic approximation on the accuracy of calculations has been of considerable interest to persons modeling acoustic propagation [see McDaniel, 1975;Thomson and Chapman, 1983, and references therein]. The parabolic wave equation obtained below for tropospheric propagation is exactly analogous to that obtained for acoustic propagation in the ocean.…”

“…In certain classes of acoustic problems the use of (22) has been found to result in significant modal phase velocity errors for propagation in directions that are more than --• 15 ø from the local horizontal [McDaniel, 1975]. The effect of these errors becomes evident in problems where an oceanic duct or shallow region gives rise to a coherent sum of several trapped modes that are propagating at relatively large angles.…”

Three-dimensional wide-angle azimuthal PE solution to modified benchmark problems

“…Next we follow McDaniel [20] by considering a single normal mode propagation in the PE approximation and comparing the results with the solution of Helmholtz equation (23).…”

“…Energy outside the angle of propagation is neglected, therefore the acoustic solution inaccurately propagated which leaded to phase errors accumulated over ranges. Phase errors in parabolic approximations were analyzed with normal-mode theory [20] and also investigated by Tappert et al [21]. However, these results only considered the vertical propagation angle.…”

10.14209/SBRT.2018.12

“…Pelo menos duas soluções numéricas podem ser usadas para resolver a PE: as técnicas de diferenças finitas [6] e o algoritmo de SSFT (Split Step Fourier Transform) [7]. Para garantir a estabilidade do algoritmo de diferenças finitas, tanto Claerbout quanto Popov usaram a solução numérica baseada no Método de Crank-Nicolson.…”